On the other hand in the classical theory it was known that the characteristic classes
can be constructed not just in terms of the usual connection and curvature procedure,
but also by simplicial cohomological methods involving \(\mathop {\rm GL}\nolimits (n, {\mathbb {R}})\)  as structure group
(cf.  [1]}, [2]}, [3]}, [4]}, [5]}).
A. Gorokhovsky figured out how to convert that kind of representation
into the formalism of Hopf cyclic cohomology by extending it to differential graded
(DG) Hopf algebras. He showed [6]}
that the suitably truncated Hopf cyclic cohomology \(HP^\bullet ((\Omega ^*\mathop {\rm GL}\nolimits (n, {\mathbb {R}}))_m)\) 
of the DG Hopf algebra of differential forms on \(\mathop {\rm GL}\nolimits (n, {\mathbb {R}})\)  captures
the characteristic classes of flat \(\Gamma \) -equivariant vector bundles of rank \(n\) , where \(\Gamma \)  is a discrete group of diffeomorphisms of the base manifold of dimension \(m\) .
In particular his results yield an isomorphism
\( HP^\bullet ({\mathcal {H}}_n, {\rm O}_n; {\mathbb {C}}_\delta ) \xrightarrow{} HP^\bullet ((\Omega ^*\mathop {\rm GL}\nolimits (n, {\mathbb {R}}))_n) ,\) 
